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2000 Q6
\(\begin{align} \text{Premium} = &\sum\limits_{i \in class} \dfrac{Payroll_i}{100} \text{Manual Rate}_i &\cdots (1)\\ &\times \: \text{Exp Mod} &\cdots (2)\\ &\times \: \text{Sch Mod} &\cdots (3)\\ &\times \: (1 - \text{Premium Discount %}) &\cdots (4) \end{align}\)
\((1)\) Manual Premium
\((1) \times (2)\) Modified Premium
\((1) \times (2) \times (3)\) Standard Premium
\((1) \times (2) \times (3) \times (4)\) Guaranteed Cost Premium
Uses insured’s loss experience during the current policy term to determine their current policy premium
Initial premium collected at the start (deposit premium) \(\Rightarrow\) Adjust based on reported loss starting 6 months after policy expires, then every 12 months after
\(\text{Premium} = \left[\text{U/W Expense & Profit xTax} + \left(1 + \dfrac{\mathrm{E}[LAE]}{\mathrm{E}[Loss]}\right) \times Loss_{actual}\right] \times \text{Tax Multiplier}\)
There’s always going to be a max and min premium or else this would defeat the purpose of risk transfer
Caps are applied to the premium amount \(\equiv\) Capping actual losses at max/min aggregate loss limits \(\Rightarrow\) Add a charge to the premium calculation to account for this
In addition to aggregate caps, can also limit losses per occurrence (Use limited losses in the retro premium calculation) \(\Rightarrow\) A separate charge to the premium calculation to account for the per occurrence limit
With both occurrence and agg limits, need to consider the overlap between the 2 charges
NCCI retro premium formula for each individual policy
\(R = (b + C \times A + P \times C \times V) \times T\) Memorize
\(R\): Retro premium @ t \(\in [G, H]\)
\(b\): Basic premium = Profit + Charge for max/min premium + non-LAE expense xTax
\(A\): Reported loss @ t (may include ALAE)
\(C\): Loss conversion factor \(= 1 + \dfrac{\mathrm{E}[LAE]}{\mathrm{E}[Loss]}\)
\(V\): Retro development factor (Optional)
\(= (1-\frac{1}{CDF}) \Rightarrow \text{Unreported Losses}\)
\(C \times A + P \times C \times V \equiv \text{B-F Ultimate}\)
To stabilize premium adjustments
Only used for the first 3 adjustments
\(P\): Standard Premium
For a balanced plan:
\(\mathrm{E}[R] = \text{Premium for Prospectively Rated Policy} = b = e - (C - 1)\mathrm{E}[A] + C \times I\) Memorize
2001 Q32
Ignore the retro development for now
\(R = (b + C \times A) \times T \: \: \: \text{for} \: H \leq R \leq G\)
Max and min premiums \(\equiv\) Aggregate limits on losses
Formula
2001 Q31
\(\begin{align} H &= (b + C \times A_H) \times T &= \text{min}\\ G &= (b + C \times A_G) \times T &= \text{max}\\ \end{align}\)
\(R = (b + C \times {\color{blue}L}) \times T \: \: \: \: {\color{blue}L} = \left\{ \begin{array} AA_H & A \leq A_H\\ A & A_H < A <A_G\\ A_G & A \geq A_G\\ \end{array} \right.\)
Understand Formula
\(\begin{array}{ccc} \text{Net Insurance Charge} = &I = (&\text{Ins Chg} &- &\text{Ins Saving}&) \times \mathrm{E}[A]\\ &I = (&\dfrac{\mathrm{E}[Loss > A_G]}{\mathrm{E}[A]} &- &\dfrac{\mathrm{E}[Loss < A_H]}{\mathrm{E}[A]}&) \times \mathrm{E}[A]\\ &I = (&\text{Tbl M Max Prem Chg} &- &\text{Tbl M Min Prem Savings}&) \times \mathrm{E}[A]\\ &I = (&\phi(r_G) &- &\psi(r_H)&) \times \mathrm{E}[A]\\ &I = (&\phi(\dfrac{A_G}{\mathrm{E}[A]}) &- &\psi(\dfrac{A_H}{\mathrm{E}[A]})&) \times \mathrm{E}[A]\\ \end{array}\)
Rows are entry ratios = \(r = \dfrac{Loss_{Actual}}{Loss_{Expected}}\) Understand
Columns are for different risk size groups
Entry ratio at maximum premium \(= r_G = \dfrac{A_G}{\mathrm{E}[A]}\)
Entry ratio at minimum premium \(= r_H = \dfrac{A_H}{\mathrm{E}[A]}\)
Per occurrence limit impact the likelihood of hitting the max/min premiums \(\Rightarrow\) Table M not appropriate \(\Rightarrow\) Need to recognize the potential overlap between the occurrence and aggregate limits
3 options to handle this:
\(R^* = (b^{LLM} + C \times A^* + P \times C \times F + P \times C \times V) \times T\)Memorize
\(b^{LLM}\): From the limited loss Table M
\(A^*\): Reported limited loss @ t (might include ALAE)
\(A \neq A^* + PF\)
\(\mathrm{E}[A] = \mathrm{E}[A^*] = PF\)
\(PCF\) and \(PCV\) are electives
\(V\): Development different with occ limit than w/o
Theoretically correct approach
Caveat:
Requires a large number of tables
Charge need to vary by occ limit, entry ratio, risk size group, HG, etc
ICRLL = Insurance Charge Reflecting Loss Limitation
Change the Table M column used to be the column appropriate for a larger size risk
Caveat:
\(R^* = (b^L + C \times A^* + P \times C \times V) \times T\)
Accurately correct for overlap between the occurrence and aggregate charges similar to the LLM
Combines occurrence charge with the aggregate charge
WCIRB uses different HG than NCCI
\(V\) varies by occ limit
NCCI do full rates or loss costs depending on the state
Full Rates:
Files \(ELF\) from option 1
Files \(V\), Retro Development Factors, from option 1
Loss Costs:
Files \(ELPPF\) (XS Loss Pure Premium Factors)
\(\begin{array}{cccc} ELF &= &ELPPF &\times &ELR\\ \dfrac{\mathrm{E}[XSLoss]}{\text{Standard Premium}} &= &\dfrac{\mathrm{E}[XSLoss]}{\mathrm{E}[Loss]} &\times &\dfrac{\mathrm{E}[Loss]}{\text{Standard Premium}}\\ \end{array}\)
\(ELR\) from company’s own estimate
Use \(ELAEPPF\) if per occ limit includes Loss & ALAE
Files Development Retrospective Pure Premium Factors
\(\begin{array}{cccc} \text{Retro Dev Factor} &= &\text{Retro Dev PP Factor} &\times &ELR\\ \dfrac{\mathrm{E}[\text{Unreported Limited Loss}]}{\text{Standard Premium}} &= &\dfrac{\mathrm{E}[\text{Unreported Limited Loss}]}{\mathrm{E}[Loss]} &\times &\dfrac{\mathrm{E}[Loss]}{\text{Standard Premium}}\\ \end{array}\)
XS Loss and Allocated Expense Pure Premium Factors
Aggregate distn across a large number of risks
Table M Charge
Revisit! This seems wrong…
\(\begin{align} \text{Table M Charge} &= \dfrac{\sum_{r_i > r} Loss_i}{\sum_i Loss_i}\\ &= \dfrac{\text{Total Losses }\forall \text{ Policies w/ Losses > Entry Ratio r}}{\text{Total Losses }\forall \text{ Policies}} \end{align}\)
\(\begin{align} \text{Table M Savings} &= \dfrac{\sum_{r_i < r} Loss_i}{\sum_i Loss_i}\\ &= \dfrac{\text{Total Losses }\forall \text{ Policies w/ Losses < Entry Ratio r}}{\text{Total Losses }\forall \text{ Policies}} \end{align}\)
2 Components to build Table M
Normalize table if the sample average \(\neq\) assumed expected value:
| Annual Claim Total ($) | # of Policies |
|---|---|
| 1000 | 8 |
| 1500 | 3 |
| 4000 | 2 |
| 10000 | 5 |
| 15000 | 2 |
| 40000 | 1 |
Each risk has $10,000 of standard premium
\(\text{Max Premium} \Leftrightarrow\) 80% LR w.r.t. Standard Premium
\(\text{Min Premium} \Leftrightarrow\) 20% LR w.r.t. Standard Premium
Loss @ Max Premium = 80% \(\times\) $10,000 = $8,000 = \(A_G\)
Loss @ Min Premium = 20% \(\times\) $10,000 = $2,000 = \(A_H\)
Total # of Policies = 21
Total Losses = $140,500
\(\mathrm{E}[A] = \text{Expected Loss per Policy} = \dfrac{\text{Total Losses}}{\text{Total # of Policies}} =\) $6,690.48 = Total area under curve
# Transformation of the data to plot
d1_ecdf <-
d1 %>%
rbind(c(0,0),.) %>%
mutate(`% Policies` = `# of Policies` / sum(`# of Policies`),
`Claim EWCDF` = cumsum(`% Policies`))
# Note the addition of 0 row
| Annual Claim Total ($) | # of Policies | % Policies | Claim EWCDF |
|---|---|---|---|
| 0 | 0 | 0.0000000 | 0.0000000 |
| 1000 | 8 | 0.3809524 | 0.3809524 |
| 1500 | 3 | 0.1428571 | 0.5238095 |
| 4000 | 2 | 0.0952381 | 0.6190476 |
| 10000 | 5 | 0.2380952 | 0.8571429 |
| 15000 | 2 | 0.0952381 | 0.9523810 |
| 40000 | 1 | 0.0476190 | 1.0000000 |
Note that the steps are drawn in a “vh” manner starting from origin
Table M Charge in dollars = Area under curve and above $8,000 line \(A_G\)
Table M Savings in dollars = Area under $8,000 line \(A_G\) and above curve
Net Insurance Charge = Difference of the Table M Charge in dollars and Table M Savings in dollars
Calculating the entry ratio first
Starting from the top of the graph
Cumulate the horizontal length of each section from top to bottom
Advantages:
1. Start with table of sampled loss amount and respective frequency
| Annual Claim Total ($) | # of Policies |
|---|---|
| 1000 | 8 |
| 1500 | 3 |
| 4000 | 2 |
| 10000 | 5 |
| 15000 | 2 |
| 40000 | 1 |
2. Add rows for 0 loss and entry ratio
tbl_m <-
d1 %>%
rbind(c(0,0), c(minlr * sp,0), c(maxlr *sp,0)) %>%
arrange(`Annual Claim Total ($)`) %>%
rename(`# risks` = `# of Policies`)
kable(tbl_m, align = 'l')
| Annual Claim Total ($) | # risks |
|---|---|
| 0 | 0 |
| 1000 | 8 |
| 1500 | 3 |
| 2000 | 0 |
| 4000 | 2 |
| 8000 | 0 |
| 10000 | 5 |
| 15000 | 2 |
| 40000 | 1 |
3. Calculate Entry Ratio \(r\) * $
tbl_m <-
tbl_m %>%
mutate(`Entry Ratio (r)` = `Annual Claim Total ($)` / weighted.mean(`Annual Claim Total ($)`, `# risks`))
kable(tbl_m, align = 'l')
| Annual Claim Total ($) | # risks | Entry Ratio (r) |
|---|---|---|
| 0 | 0 | 0.0000000 |
| 1000 | 8 | 0.1494662 |
| 1500 | 3 | 0.2241993 |
| 2000 | 0 | 0.2989324 |
| 4000 | 2 | 0.5978648 |
| 8000 | 0 | 1.1957295 |
| 10000 | 5 | 1.4946619 |
| 15000 | 2 | 2.2419929 |
| 40000 | 1 | 5.9786477 |
4. Calculate # of risk above each row (Loss)
tbl_m <-
tbl_m %>%
mutate(`# risks above` = sum(`# risks`) - cumsum(`# risks`))
kable(tbl_m, align = 'l')
| Annual Claim Total ($) | # risks | Entry Ratio (r) | # risks above |
|---|---|---|---|
| 0 | 0 | 0.0000000 | 21 |
| 1000 | 8 | 0.1494662 | 13 |
| 1500 | 3 | 0.2241993 | 10 |
| 2000 | 0 | 0.2989324 | 10 |
| 4000 | 2 | 0.5978648 | 8 |
| 8000 | 0 | 1.1957295 | 8 |
| 10000 | 5 | 1.4946619 | 3 |
| 15000 | 2 | 2.2419929 | 1 |
| 40000 | 1 | 5.9786477 | 0 |
5. Calculate % of risk above each row
tbl_m <-
tbl_m %>%
mutate(`% risks above` = `# risks above` / sum(`# risks`))
kable(tbl_m, align = 'l')
| Annual Claim Total ($) | # risks | Entry Ratio (r) | # risks above | % risks above |
|---|---|---|---|---|
| 0 | 0 | 0.0000000 | 21 | 1.0000000 |
| 1000 | 8 | 0.1494662 | 13 | 0.6190476 |
| 1500 | 3 | 0.2241993 | 10 | 0.4761905 |
| 2000 | 0 | 0.2989324 | 10 | 0.4761905 |
| 4000 | 2 | 0.5978648 | 8 | 0.3809524 |
| 8000 | 0 | 1.1957295 | 8 | 0.3809524 |
| 10000 | 5 | 1.4946619 | 3 | 0.1428571 |
| 15000 | 2 | 2.2419929 | 1 | 0.0476190 |
| 40000 | 1 | 5.9786477 | 0 | 0.0000000 |
6. Calculate the Table M Charge \(\phi(r)\)
\(\phi(r) = 0\) for the largest value of r \(\Rightarrow\) No \(\mathrm{E}[L]\) above the largest loss
\(\phi(r_i) = \phi(r_{i+1}) + (r_{i+1} - r_i)(\text{ % risks above }r_i)\) Memorize Formula
Savings \(= \psi(r) = \phi(r) + r - 1\) Memorize Formula
Looking at groups of losses based on the size of loss
Probably start from the right but don’t seem to matter
Advantages:
Note that the y axis is now the entry ratios and not losses like it was in the earlier example
Defn:
\(Y = \dfrac{A}{\mathrm{E}[A]}\)
\(F(y) = \text{CDF of } y\)
Area under curve:
\(\mathrm{E}[Y] = \dfrac{\mathrm{E}[A]}{\mathrm{E}[A]} = 1\)
Table M Charge and Savings Formulas
\(\phi(r) = \int\limits_r^{\infty} (y - r) dF(y)\)
\(\psi(r) = \int\limits_r^{\infty} (r - y) dF(y)\)
Key Properties - Tbl M Charge Important
\(\phi(0) = 1, \: \phi(\infty) = 0\)
\(\phi(r)'(f) = -G(r) \leq 0\)
\(\phi''(r) = f(r)\)
Key Properties - Tbl M Savings Important
\(\psi(0) = 0, \: \psi(\infty) = \infty\)
\(\psi(r)'(f) = F(r) \geq 0\)
\(\psi''(r) = f(r) = \phi''(r)\)
Note from figure above, the area under \(r = \psi(r) + (1 - \phi(r)) \Rightarrow \psi(r) = \phi(r) + r -1\)
\(\phi\) is monotonically decreasing function of premium size
\(R = (b + C \times {\color{blue}L}) \times T \: \: \: \: {\color{blue}L} = \left\{ \begin{array} AA_H & A \leq A_H\\ A & A_H < A <A_G\\ A_G & A \geq A_G\\ \end{array} \right.\)
\(A_H = r_H \mathrm{E}[A]\) and similarly for \(A_G\)
Blue area = \(\dfrac{\mathrm{E}[{\color{blue}L}]}{\mathrm{E}[A]}\)
\(\begin{array}{ll} \dfrac{\mathrm{E}[L]}{\mathrm{E}[A]} &= 1 + \psi(r_H) - \phi(r_G) \\ \mathrm{E}[L] &= \mathrm{E}[A] + [\psi(r_H) - \phi(r_G)]\mathrm{E}[A] \\ \mathrm{E}[L] &= \mathrm{E}[A] - I \end{array}\)
Memorize Understand
\(\begin{array}{lll} R &= (b + CL)T\\ \mathrm{E}[R] &= (b + C\mathrm{E}[L])T\\ \mathrm{E}[R] &= (b + C(\mathrm{E}[A] - I))T & (1)\\ \end{array}\)
Recall if the plan is balance \(\Rightarrow\) \(\mathrm{E}[R] = \text{Guaranteed Cost Premium}\)
\(\begin{array}{lll} \text{Guaranteed Cost Premium} &= (e + \mathrm{E}[A])T & (2)\\ &=(1-D)P & (3)\\ \end{array}\)
Combine \((1)\) and \((2)\)
\((b + C(\mathrm{E}[A] - I))T = (e + \mathrm{E}[A])T\)
\(b = e - (C - 1)\mathrm{E}[A] + CI\) Memorize Formula
2009 Q31, 2011 Q8
Question that ask for G or H usually use balance equation
Alt formula: \(b = (1 - D)\dfrac{P}{T} - C\mathrm{E}[A] + CI\)
Use this if we don’t’ have max/min premium
Basic premium and max/min premium depends on each other \(\Rightarrow\) Need trial and error to get the right Table M row
Value (change) difference
Retro premium formula with max loss:
\(H = (b + CA_H)T = (b + Cr_H\mathrm{E}[A])T\)
Start with the GCP formula and subtract \(H\)
\(\begin{array}{ll} \text{GCP} - {color{blue}H} &\\ (e + \mathrm{E}[A])T - {\color{blue}H} &= (b + C(\mathrm{E}[A] - I))T - {\color{blue}{(b + Cr_H\mathrm{E}[A])T}}\\ &= CT\{\mathrm{E}[A] - I - r_H\mathrm{E}[A]\}\\ &= CT\{\mathrm{E}[A] - \mathrm{E}[A](\phi(r_G) - \psi(r_H)) - r_H \mathrm{E}[A]\}\\ &= C\mathrm{E}[A]T(\psi(r_H) - r_H + 1 - \phi(r_G))\\ \end{array}\)
Move everything to one side and we get: Important Memorize
\(\phi(r_H) - \phi(r_G) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A]T} = \dfrac{(1-D)P-H}{C\mathrm{E}[A]T}\)
Entry difference, based \(G - H\)
Important Memorize
\(r_G - r_H = \dfrac{G - H}{C\mathrm{E}[A]T}\)
Can do all the above looking at them as a % of Standard Premium
Note:
Same as Table M but with limited losses \(A^*\) for each policy
Entry Ratio: \(r^* = \dfrac{A^*}{\mathrm{E}[A^*]}\)
Note that the denominator of the Entry Ratio is the \(\mathrm{E}[A^*]\), where as in Table L it’s not limited \(\mathrm{E}[A^*]\)
Note that the y axis is now limited entry ratios \(y^*\)
Entry Ratio: \(Y^* = \dfrac{A^*}{\mathrm{E}[A^*]}\)
\(F^*(Y^*) = \text{CDF of }Y^*\)
Loss Elimination Ratio: \(k = LER = 1 - \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]}\) Know
Note that \(\mathrm{E}[Y^*] = \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A^*]}\) no surprise
But \(\mathrm{E}[Y] = \dfrac{\mathrm{E}[A]}{\mathrm{E}[A^*]} = 1 + k\dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]}\)
Integration and key properties all the same as Table M but with limited loss
Same thing for what happens when \(\lim\limits_{\text{Premium Size} \rightarrow \infty}\) and \(\lim\limits_{\text{Premium Size} \rightarrow 0}\)
\(R^* = (b^{LLM} + C{\color{blue}{L^*}}+PCF) T \: \: \: \: {\color{blue}{L^*}} = \left\{ \begin{array} AA_H & A^* \leq A_H\\ A^* & A_H < A^* <A_G\\ A_G & A^* \geq A_G\\ \end{array} \right.\)
\(+ PCV\) if to stabilize
Everything similar to Table M but now that the entry ratios \(r^*\) are limited
\(\mathrm{E}[L^*] = \mathrm{E}[A^*] - I^{LLM}\)
\(b^{LLM} = e - (C - 1)\mathrm{E}[A] + CI^{LLM}\) * \(e - (C - 1)\mathrm{E}[A]\) is never capped
\(\phi^{LLM}(r_H^*) - \phi^{LLM}(r_G^*) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A^*]T} = ? \dfrac{(1-D)P-H}{C\mathrm{E}[A^*]T}\)
\(r_G^* - r_H^* = \dfrac{G - H}{C\mathrm{E}[A^*]T}\)
Table L charge includes the charge for per occurrence limit
Implicit charge for the occ limit \(\in \{0, k\}\)
Table L uses different premium size groups and occurrence limits
The LER was derived from the combination of all premium groups
CA version of Table M shows a higher charge than the CW average due to higher variation in the CA rates
\(\phi^L(r) - \phi(r)\) shows in the data was smaller than the LER implies due to overlap between occ limit and agg limit
Main Difference from the M and LLM
Entry ratios = \(\dfrac{A^*}{\mathrm{E}[A]} = \dfrac{\text{Actual Limited Loss}}{\text{Expected }\textbf{Unlimited}\text{ Loss}}\)
2013 Q15b
If Standard Premium \(\neq\) \(\forall\) risk \(\Rightarrow\) Create Tbl L with P instead of # of risk
2008 Q32
Note that the y axis is \(Y = \dfrac{A^*}{\mathrm{E}[A]}\) and the CDF of interest is \(F^*(Y)\). However, the x axis of the graph is of \(F(Y)\)
Loss Elimination Ratio: \(k = LER = 1 - \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]}\)
Area under \(F(Y) = 1\) same as Table M \(\Rightarrow\) Area under \(F^*(Y) = \mathrm{E}[Y] = \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]} = 1 - k\)
\(\phi^L(r) = \int\limits_r^{\infty} (y-r)dF^*(y) + k\) Remember the k
\(\psi^L(r) = \int\limits_0^{\r} (r-y)dF^*(y)\)
Key properties all the same except:
\(\phi^L(\infty) = k\)
\(R^* = (b^L + C{\color{blue}{L^*}}) T \: \: \: \: {\color{blue}{L^*}} = \left\{ \begin{array} AA_H & A^* \leq A_H\\ A^* & A_H < A^* <A_G\\ A_G & A^* \geq A_G\\ \end{array} \right.\)
Everything same as Table M but now that we use \(I^L\)
\(\mathrm{E}[L^*] = \mathrm{E}[A^*] - I^{L}\)
\(b^{L} = e - (C - 1)\mathrm{E}[A] + CI^{L}\) * \(e - (C - 1)\mathrm{E}[A]\) is never capped
\(\phi^{L}(r_H) - \phi^{L}(r_G) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A]T} = ? \dfrac{(1-D)P-H}{C\mathrm{E}[A]T}\)
\(r_G - r_H = \dfrac{G - H}{C\mathrm{E}[A]T}\)
2001 Q34
NCCI uses the ICRLL to \(\approx\) LLM
Table L Advantages:
Table L Disadvantages:
See Manual for example
Some of the important sections below. Cancellation provision is not on the syllabus
Expected Loss Group = Risk Size Groups
To determine the ELG:
\(\mathrm{E}[L]\) increase over time due to inflation. The ELG tables are updated so the curves so NCCI doesn’t have to update the curve. They’ll just push risk to different ELG.
Estimate portion of risk moving from ELG X to Y: \(\dfrac{\text{Portion of ELG X in new ELG Y}}{\text{Size of old ELG X}}\)
Each column is for different ELG
For low entry ratios, Table M savings are shown as well
Not tested since 2000
This is for the \(e\) in \(b = e - (C - 1)\mathrm{E}[A] + CI\)
Type A = Stock; Type B = non-Stock
Shows ELR, premium discount ranges and tax multipliers
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